#OK to post homework
#Joseph Koutsoutis, 04-07-2024, Assignment 20

read `C21.txt`:
with(plots):


# For this assignment, I changed the last line of our AddE function
# to return the additive inverse of the line through P and Q since our
# AddE from lecture 20 always returns a non-positive y value.
AddE:=proc(a,b,P,Q) local x,y,s,eq,sol,R,xR:
 s:=(P[2]-Q[2])/(P[1]-Q[1]):
  y:=P[2]+(x-P[1])*s:
sol:={solve(y^2-x^3-a*x-b,x)};


if not (nops(sol)=3 and member(P[1],sol) and member(Q[1],sol)) then
  print(`Something bad happened`):
  RETURN(FAIL):
fi:

sol:=sol minus {P[1],Q[1]}:
xR:=sol[1]:
[xR,-(P[2]+(xR-P[1])*s)]: # this is the only line changed

end: 


#1 

Diag := proc(a,b) local A,B,C,A_label,B_label,C_label,AB_label,ecplot,points:
 A := RP(a,b):
 B := RP(a,b):
 while A[1] = B[1] do:
  B := RP(a,b):
 od:
 C := AddE(a,b,A,B):

 A_label := textplot([op(A), "A"]):
 B_label := textplot([op(B), "B"]):
 C_label := textplot([C[1],-C[2], "C"]):
 AB_label := textplot([op(C), "A+B"]):
 points := pointplot([A,B,[C[1],-C[2]],C], connect=true):
 ecplot := implicitplot(y^2 = x^3 + a*x + b):

 display([A_label, B_label, C_label,AB_label, points,ecplot]):
end:


#2

IntSol := proc(a,b,K) local x,y,S:
 S := {}:
 for x from -K to K do:
  y := x^3 + a*x + b:
  if y >= 0 then:
   y := sqrt(y):
   if trunc(y) = y then:
    S := S union {[x,trunc(y)], [x,-trunc(y)]}:
   fi:
  fi:
 od:
 S:
end: 


#3

SolChain := proc(a,b,S1,S2,K) local L:
 L := [S1,S2]:
 while nops(L) < K+2 do:
  L := [op(L), AE(a,b,L[-2],L[-1])]:
 od:
 L:
end:


# SolChain ended up taking a long time to run so I started
# using floats instead of keeping track of exact values for
# testing larger K.
SolChainFloat := proc(a,b,S1,S2,K) local L:
 L := [S1,S2]:
 while nops(L) < K+2 do:
  L := [op(L), evalf(AE(a,b,L[-2],L[-1]))]:
 od:
 L:
end:


# I didn't test too many values but I couldn't find two intial solutions
# that gave an arbitrarily long chain of solutions.